Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.7 Theoretical and computing modeling 481
and ρ M, and its solution is represented by a set of φvalues in the grid points. These values
have to be reached iteratively.
Many technical details of the methods have been changed in the years. We signal,
among others, the strategy used to define the portion of the grid, especially near the molecular
surface where there is a sharp change in ε, and the mathematical algorithms adopted to
improve convergence in the iterative processes. Almost all FD methods have been conceived
for, and applied to, classical descriptions of solutes, generally represented by atomic
charges. The latter are usually drawn from previous QM calculations in vacuo or from Molecular
Mechanics effective potentials, as in the study of large molecules, usually proteins.
The deficiencies due to the use of rigid charges are partially alleviated by introducing a dielectric
constant greater than one for the inner space (ε between 2 and 4), and/or by including
atomic polarization functions (akin to atomic dipole polarizabilities).
The DelPhi program 85a is one of these proposals. This method, which has been extensively
improved during the years, exists now in a QM version 86 exploiting a recent high
level QM method, called PS-GVB, 87 based on a synthesis of grid point integration techniques
and standard Gaussian orbital methods. From a PS-GVB calculation in vacuo, a set
of atomic charges, the potential derived (PD) charges, is obtained by a fitting of the molecular
electrostatic potential (MEP). These charges are used in DelPhi to solve the Poisson
equation in the medium, and then to get, with the aid of a second application of DelPhi in
vacuo, the reaction field (RF). The latter is represented by a set of point charges on the cavity
surface, which define the solute-solvent interaction potential to be inserted in the
PS-GVB Hamiltonian. Then a new computational cycle starts, and the whole process is repeated
until convergence is reached.
In parallel, the classical FD procedure of Bashford and Karplus 88 has been coupled
with the Amsterdam density fuctional theory (DFT) 89 code to give another version of
QM-FD methods. Starting from a DFT calculation in vacuo, the RF potential is obtained as
the difference between the solutions of the Poisson equation obtained with a FD method in
the medium and in vacuo (in both cases, the electrostatic potential φ M is expressed in terms
of PD atomic charges). The RF potential is then re-computed to solve a modified
Kohn-Sham equation. The calculations are repeated until convergence is achieved.
An alternative to the use of finite differences or finite elements to discretize the differential
operator is to use boundary element methods (BEM). 90 One of the most popular of
these is the polarizable continuum model (PCM) developed originally by the Pisa group of
Tomasi and co-workers. 32 The main aspect of PCM is to reduce the electrostatic Poisson
equation (1) into a boundary element problem with apparent charges (ASCs) on the solute
cavity surface.
A method that is very similar to PCM is the conductor-like screening model
(COSMO) developed by Klamt and coworker 91 and used later by Truong and coworkers, 92
and by Barone and Cossi. 93 This model assumes that the surrounding medium is well modeled
as a conductor (thus simplifying the electrostatics computations) and corrections are
made a posteriori for dielectric behavior.
Electrostatic BE methods can be supported by nonelectrostatic terms, such as dispersion
and exchange, in different ways, and their basic theory can be extended to treat both
classical and quantum mechanical solutes; in addition, many features, including analytical
gradients with respect to various parameters, have been added to the original models so as to